Stationary action

Definition

Stationary action is a principle in physics and mathematics stating that the path taken by a system is one for which the action integral is stationary (usually a minimum or saddle point) with respect to variations of the path. This principle underlies various formulations of mechanics, leading to equations that describe the motion of systems and geometric properties of paths in space.

5 Must Know Facts For Your Next Test

1. The principle of stationary action can be used to derive the Euler-Lagrange equations, which provide the conditions for a function to be extremized.
2. In the context of geodesics, stationary action implies that the shortest or extremal path between two points in a curved space-time corresponds to the actual path taken by a particle.
3. Stationary action applies not only in classical mechanics but also in fields such as quantum mechanics and general relativity, making it a universal principle across different areas of physics.
4. The concept relies on calculus of variations, where small changes in paths lead to small changes in the action integral, and one looks for paths where these changes vanish.
5. When applying stationary action, one often considers boundary conditions that specify the endpoints of the trajectory being studied.

Review Questions

• How does the principle of stationary action lead to the derivation of Euler-Lagrange equations?
  • The principle of stationary action states that for a physical system, the true path taken is one that makes the action integral stationary. By applying calculus of variations to this integral, we can derive the Euler-Lagrange equations. These equations provide a systematic way to find the equations of motion for a system by setting conditions on how small variations in path affect the action, leading to precise mathematical formulations.
• What role does stationary action play in determining geodesics in differential geometry?
  • In differential geometry, stationary action relates to geodesics as it identifies the paths that minimize or extremize distance between points on curved surfaces. When applying this principle, one can derive equations that describe these paths using variational methods. Thus, geodesics can be viewed as trajectories where changes in path do not alter the action, establishing them as natural generalizations of straight lines in Euclidean space.
• Critically evaluate how the concept of stationary action integrates with modern physics theories beyond classical mechanics.
  • The concept of stationary action serves as a foundational idea across various fields of modern physics beyond classical mechanics. In quantum mechanics, for instance, it manifests through Feynman's path integral formulation where all possible paths contribute to the probability amplitude. Additionally, in general relativity, the principle guides how spacetime curvature affects particle trajectories. This integration shows that stationary action not only unifies classical descriptions but also extends into realms like quantum field theory and cosmology, highlighting its fundamental importance in our understanding of physical laws.

Related terms

Lagrangian: The Lagrangian is a function that summarizes the dynamics of a system, representing the difference between kinetic and potential energy, and is central to deriving the equations of motion using the principle of stationary action.

Action Integral: The action integral is a quantity obtained by integrating the Lagrangian over time, serving as a functional whose stationary points correspond to the actual paths taken by physical systems.

Variational Principle: The variational principle is a fundamental concept in physics that posits that certain quantities (like action) reach extrema (minimum, maximum, or saddle points) under specific conditions, leading to the derivation of equations governing physical behavior.